Standard half-cell potentials

When a net reaction proceeds in an electrochemical cell, oxidation occurs at one electrode (the anode) and reduction takes place at the other electrode (the cathode.) We can think of the cell as consisting of two half-cells joined together by an external circuit through which electrons flow and an internal pathway that allows ions to migrate between them so as to preserve electroneutrality.

Reduction potentials

Each half-cell has associated with it an electrode-solution potential difference whose magnitude depends on the nature of the particular electrode reaction and on the concentrations of the dissolved electroactive species. The sign of this potential difference depends on the direction (oxidation or reduction) in which the electrode reaction proceeds. In order express them in a uniform way, we adopt the convention that half-cell potentials are always defined for the reduction direction. Thus the half-cell potential for the Zn/Zn²⁺ electrode (or couple as it is sometimes called) always refers to the reduction reaction

Zn²⁺ + 2e^– → Zn(s)

In the cell Zn(s) | Zn²⁺(aq) || Cu²⁺(aq) | Cu(s) the zinc appears on the left side, indicating that it is being oxidized, not reduced. For this reason, the potential difference contributed by the left half-cell has the opposite sign to its conventional half-cell potential. More generally, we can define the cell potential or cell EMF as

E_cell = ΔV = E_right – E_left            (1)

in which "right" and "left" refer to the cell notation convention ("reduction on the right") and not, of course, to the physical orientation of a real cell in the laboratory. If we expand the above expression we see that the cell potential

E_cell = V_Cu – V_solution + V_solution – V_Zn

is just the difference between the two half-cell potentials E_right and E_left.

Reference half-cells

The fact that individual half-cell potentials are not directly measurable does not prevent us from defining and working with them. Although we cannot determine the absolute value of a half-cell potential, we can still measure its value in relation to the potentials of other half cells. In particular, if we adopt a reference half-cell whose potential is arbitrarily defined as zero, and measure the potentials of various other electrode systems against this reference cell, we are in effect measuring the half-cell potentials on a scale that is relative to the potential of the reference cell.

The reference cell that has universally been adopted for this purpose is the hydrogen half-cell

Pt | ½ H₂(g) | H⁺(aq) || ...

in which hydrogen gas is allowed to bubble over a platinum electrode having a specially treated surface which catalyzes the reaction

½ H₂(g) → H⁺ + e^–

When this electrode is operated under standard conditions of 1 atm H₂ pressure, 25°C, and pH = 0, it becomes the standard hydrogen electrode, sometimes abbreviated SHE.

In order to measure the relative potential of some other electrode couple M²⁺/M, we can set up a cell

Pt | H₂(g) | H⁺ || M²⁺ (aq) | M(s)

whose net reaction is

H₂(g) + M²⁺(aq) → 2H⁺ + M(s)

the potential difference between the platinum and M electrodes will be

E_cell = V_M – V_solution + V_solution – V _Pt

but since the difference V_solution – V _Pt is by definition zero for the hydrogen half-cell, the cell potential we measure corresponds to

E_cell = V_M – V_solution

which is just the potential (relative to that of the SHE) of the half-cell whose reaction is

M²⁺ + 2e^– → M(s)

Measurement of a standard reduction potential. The M2+/M half-cell is on the left, and the standard hydrogen cell is on the right. The two half-cells are joined through the salt bridge in the middle. The more "active" the metal M (the greater its tendency to donate electrons to H+), the more negative will be Ecell = ΔV = Eright – Eleft

Standard [reduction] potentials are commonly denoted by the symbol E°. E° values for hundreds of electrodes have been determined (mostly in the period 1925-45, during which time they were referred to as "oxidation potentials") and are usually tabulated in order of increasing tendency to accept electrons (increasing oxidizing power.)

Cell potentials and Free (Gibbs) energy

From the above, it should be apparent that the potential difference between the electrodes of a cell is a measure of the tendency for the cell reaction to take place: the more positive the cell potential, the greater the tendency for the reaction to proceed to the right. But we already know that the standard free energy change expresses the tendency for any kind of process to occur under the conditions of constant temperature and pressure. Thus ΔG° and E° measure the same thing, and are related in a simple way:

... or in more detail (see below for explanations of the units given for voltage)

A few remarks are in order about this very fundamental and important relation:

• The negative sign on the right indicates that a positive cell potential (according to the sign convention discussed previously) implies a negative free energy change, and thus that the cell reaction will spontaneously proceed to the right.
• Electrical work is done when an electric charge q moves through a potential difference ΔV. The right side of Eq. 2 refers to the movement of n moles of charge across the cell potential E°, and thus has the dimensions of work.
• The value of ΔG° expresses the maximum useful work that a system can do on the surroundings. "Useful" work is that which can be extracted from the cell by electrical means to operate a lamp or some other external device. This excludes any P-V work that is simply a consequence of volume change (which could conceivably be put to some use!) and which would be performed in any case, even if the reactants were combined directly. This quantity of work –ΔG° can only be extracted from the system under the limiting conditions of a thermodynamically reversible change, which for an electrochemical cell implies zero current. The more rapidly the cell operates, the less electrical work it can supply.
• If F is expressed in coulombs per mole of electrons, the electrical work is in joules per mole. To relate these units to electrical units, recall that the coulomb is one amp-sec, and that power, which is the rate at which work is done, is measured in watts, which is the product of amps and volts

1 J = 1 watt-sec = 1 (amp-sec) × volts

If Eq. 2 is solved for E°, we have

(3)

This states explicitly that the cell potential is a measure of the free energy change per mole of electrons transferred, which is a brief re-statement of the principle explained immediately above. To see this more clearly, consider the cell

Cu(s) | Cu²⁺ || Cl^– | AgCl(s) | Ag(s)

for which we list the standard reduction potentials and ΔG°s of the half-reactions:

Here we multiply the cathodic reaction by two in order to balance the charge. Because the anodic reaction is written as an oxidation, we reverse the sign of its E° and obtain E_cell = E_right – E_left = –.115 volt for the cell potential. The negative cell potential tells us that this reaction will not proceed spontaneously.

When the electrons don't cancel out

Note, however, that if we are combining two half reactions to obtain a third half reaction, the E° values are not additive, since this third half-reaction is not accompanied by another half reaction that causes the charges to cancel. Free energies are always additive, so we combine them, and use ΔG° =  –nFE° to find the cell potential.

The fall of the electron

A table of standard half-cell potentials summarizes a large amount of chemistry, for it expresses the relative powers of various substances to donate and accept electrons by listing reduction half-reactions in order of increasing E° values, and thus of increasing spontaneity. The greater the value of E°, the greater the tendency of the substance on the left to acquire electrons, and thus the stronger this substance is as an oxidizing agent.

If you have studied elementary chemical thermodynamics, you will have learned about the role that a quantity called the Gibbs free energy, usually referred to as simply the "free energy", plays in determining the direction of any chemical change. The rule is that all spontaneous change (that is, all reactions that proceed to the "right") is associated with a fall in the free energy, and the greater the degree of that fall (ΔG°), the greater will be the tendency for the reaction to take place.

If you are not familiar with the concept of free energy, just think of it as something like potential energy, which similarly decreases when spontaneous mechanical events occur, such as the dropping of a weight.

Since oxidation-reduction processes involve the transfer of an electron from a donor to an acceptor, it makes sense to focus on the electron and to consider that it falls from a higher-free energy environment (the reductant, or "source") to a lower-free energy one (the oxidant, or "sink".)

As can be seen from the diagram below, this model makes it far easier to predict what will happen when two or more oxidants and reducants are combined; the electron "falls" as far as it can, filling up oxidizing agents (sinks) from the bottom up, very much in the same way as electrons fill atomic orbitals as we build up larger atoms.